Method and computer program product for generating an artefact reduced voxel data record

ABSTRACT

The present invention relates to a method and a computer program product for generating an artifact-reduced voxel data record of an object. The artifact-reduced voxel data record of the object is generated with the aid of a computed tomography scanner. In some aspects of the present disclosure, a first image data record and a second image data record is generated by acquiring computed tomography images of the object. In still other aspects of the present disclosure, the artifact-reduced voxel data record of the object is generated with the aid of an image data reconstruction algorithm.

RELATED APPLICATIONS

The present application is a U.S. non-provisional filing of GermanPatent Application No. 10 2015 007 934.4, filed on Jun. 19, 2015, andthe present application claims priority to and the benefit of theabove-identified application, which is incorporated by reference hereinin its entirety.

BACKGROUND

The invention relates to a method for generating an artefact-reducedvoxel data record of an object to be examined, with the aid of acomputed tomography scanner and to a computer program product in thisrespect.

X-ray computed tomography (CT) is a method for obtaining informationabout the interior of objects. Computed tomography originates from themedical field but, in the meantime, it is also used in the industrialfield for material analysis and for non-destructive examinations.

In x-ray computed tomography, artefacts arise as a result of variouseffects, e.g. as a result of the employed reconstruction method and as aresult of beam hardening. The examination results can be influencedsignificantly by metallic artefacts, particularly in the case ofindustrial computed tomography, i.e. in the examination of technicalobjects such as e.g. printed circuit boards by way of computedtomography. Thus, metallic artefacts can cause e.g. streaks in thereconstructed data records and/or make the identification of structureswhich adjoin the metals more difficult or prevent the latter.Reconstruction and beam hardening artefacts also have a negativeinfluence on the quality of x-ray computed tomography data records andcan cause problems during further use of the data (e.g. in the case ofedge detection algorithms).

Previous methods for reducing artefacts are either very time andcomputationally intensive or can only correct specific parts of theobject to be examined, e.g. non-metallic parts.

It is therefore an object of the present invention to provide a methodand a computer program product, by means of which artefacts, inparticular reconstruction and beam hardening artefacts, can be reducedin computed tomography.

This object is achieved by the subject matter of the coordinate claims.Advantageous embodiments are the subject matter of the dependent claims.

A first independent aspect for achieving the object relates to a methodfor generating an artefact-reduced voxel data record of an object to beexamined, with the aid of a computed tomography scanner, comprising thefollowing steps in the specified sequence:

-   -   generating a first image data record by acquiring a multiplicity        of first computed tomography images of the object, wherein an        acquisition angle in respect of a first axis of rotation is        modified between the acquisition of the first computed        tomography images;    -   tilting the object by a predetermined tilt angle in respect of a        second axis of rotation which is arranged substantially        orthogonal to the first axis of rotation;    -   generating a second image data record by acquiring a        multiplicity of second computed tomography images of the object        tilted about the second axis of rotation;    -   generating the voxel data record of the object to be examined,        with the aid of an iterative image data reconstruction algorithm        which uses both the generated first image data record and the        generated second image data record as an input data record.

Within the meaning of this description, a three-dimensional (3D) voxeldata record or else volume data record is understood to mean a datarecord which comprises a multiplicity of voxels. Here, a voxel is a gridpoint or pixel in a three-dimensional grid or coordinate system. Hence,the multiplicity of voxels of the voxel data record represents thethree-dimensional volume of the object to be examined, in the form ofdiscrete points. The voxel data record comprises a value for each voxel,which value describes the attenuation of x-ray radiation at the locationof the voxel, i.e. at a specific three-dimensional point of the objectto be examined.

The term “acquisition” of images comprises, in particular, recording ormeasuring images.

A first image data record is generated by acquiring a multiplicity offirst computed tomography images, i.e. a first series of images or afirst image sequence, of the object. The images are acquired with theaid of an acquisition unit which comprises one or more detectors, e.g. aflat-panel detector. In particular, the first image data recordcomprises a multiplicity of first computed tomography images or a firstseries of images or a first image sequence. The individual images of thefirst image data record are in each case acquired from differentperspectives, or acquisition or recording angles. To this end, anacquisition angle is modified in respect of a first axis of rotation ofthe object or of the computed tomography scanner between the acquisitionof the first computed tomography images. By way of example, the objectcan be rotated about an axis of rotation between the acquisition of theindividual images. Alternatively or additionally, an acquisition unitcan be rotated about an axis of rotation between the acquisition of theindividual images. In particular, each image of the first image datarecord can be associated with a specific perspective or a specificacquisition angle. Preferably, the first image data record comprisesimages for acquisition angles from 0° to 180°, more preferably from 0°to 360°.

Accordingly, a second image data record is generated by acquiring amultiplicity of second computed tomography images, i.e. a second seriesof images or a second image sequence, of the object. The images areacquired with the aid of an acquisition unit which comprises one or moredetectors, e.g. a flat-panel detector. In particular, the second imagedata record comprises a multiplicity of second computed tomographyimages or a second series of images or a second image sequence. Justlike the images of the first image data record, the individual images ofthe second image data record are in each case acquired from differentperspectives, or acquisition or recording angles. To this end, theacquisition angle is modified in respect of the first axis of rotationof the object or of the computed tomography scanner between theacquisition of the second computed tomography images. By way of example,the object can be rotated about an axis of rotation between theacquisition of the individual images. Alternatively or additionally, anacquisition unit can be rotated about an axis of rotation between theacquisition of the individual images. In particular, each image of thesecond image data record can be associated with a specific perspectiveor a specific acquisition angle. Preferably, the second image datarecord comprises images for acquisition angles from 0° to 180°, morepreferably from 0° to 360°.

Between the acquisition of the first computed tomography images and theacquisition of the second computed tomography images, the object istilted by a predetermined or prescribed tilt angle in respect of asecond axis of rotation. The second axis of rotation is orientedsubstantially orthogonal to the first axis of rotation. Preferably, thesecond axis of rotation corresponds to an optical axis of the computedtomography scanner, wherein the optical axis is defined e.g. by theconnecting line between an x-ray source and the detector of the computedtomography scanner. In principle, the tilt angle can assume any valuegreater than 0° and less than 360°. By way of example, the tilt angle isapproximately 30°, 60° or 120°. Preferably, the tilt angle isapproximately 90°.

Furthermore, the first and/or second image data record preferablycomprises metadata for each acquired image, which metadata describe theposition and/or the perspective or the acquisition angle of the object.

Both the first image data record and the second image data record formthe input data record for an iterative image data reconstructionalgorithm, by means of which the artefact-reduced voxel data record ofthe object to be examined is generated or calculated.

It is also possible that, in addition to the first image data record andsecond image data record, one or more further image data records, e.g. athird, fourth, fifth, etc. image data record, is/are generated in amanner analogous to the first image data record and second image datarecord, wherein the object is tilted in relation to the second axis ofrotation, in particular by the predetermined tilt angle or by adifferent predetermined tilt angle, between the acquisition of theimages associated with the respective image data records in each case.

In the method according to the invention, a plurality of x-ray datarecords are advantageously combined during the reconstruction in orderto reduce the artefacts and in order to improve the accuracy of valueswhich are obtained from the voxel data. It was found that artefacts suchas reconstruction and beam hardening artefacts generally have adirectional dependence and, in particular, extend away from thestructures in the voxel data record in a manner orthogonal to theemployed axis of rotation.

As a result of two image data records of the object being generated orrecorded in the method according to the invention, said image datarecords differing in that the object is tilted or rotated orthogonallyin relation to the first axis of rotation, in particular by 90 degrees,between the recording of the first image data record and of the secondimage data record, the arising artefacts extend in different directions.The iterative image data reconstruction algorithm simultaneously usesboth data records of the same object in different orientations as inputand, as a result, supplies a voxel data record with significantlyreduced artefacts.

In a preferred embodiment of the method according to the invention, theiterative image data reconstruction algorithm is based on a maximumlikelihood expectation maximization (MLEM) algorithm.

In particular, the iterative image data reconstruction algorithm is amodified MLEM algorithm which is designed to use or process a pluralityof different image data records, in particular two image data records,of the object simultaneously as an input or as an input data record. Anartefact-reduced voxel data record can be calculated iteratively on thebasis of the two image data records or on the basis of the plurality ofimage data records by means of the modified MLEM algorithm.

In a further preferred embodiment of the method according to theinvention, the image data reconstruction algorithm comprises acalculation of a normalization volume data record, wherein thenormalization volume data record emerges as a sum of a normalizationvolume data record associated with the first image data record and anormalization volume data record associated with the second image datarecord.

Expressed in formulae, the normalization volume data record norm iscalculated as follows:norm=P _(rot1) ^(T)(normseq₁)+P _(rot2) ^(T)(normseq₂)  (1),whereP_(rot) ^(T)(I) generally represents a transposed projection or a backprojection of an image sequence I, which is rotated by the invertedquaternion rot. The index 1 in Equation (1) in this case means that theback projection relates to first image data record, while the index 2accordingly means that the back projection relates to the second imagedata record. In particular, rot1 is a quaternion which describes therotation of the object for the first image data record, with rot1therefore being an identical rotation, i.e. rot1:=1. Accordingly, rot2is a quaternion which describes the rotation of the object for thesecond image data record. To the extent that the tilt of the objectbetween the acquisition of the images of the first image data record andthe acquisition of the images of the second image data record is 90°about the z-axis, the following applies:

${{rot}\; 2} = {\frac{1}{\sqrt{2}} + {k{\frac{1}{\sqrt{2}}.}}}$normseq₁ means a normalized image sequence of the first image datarecord and normseq2 means a normalized image sequence of the secondimage data record.In particular,normseq₁:=1 and normseq₂:=1  (2)are set in Equation (1).

In a further preferred embodiment of the method according to theinvention, the image data reconstruction algorithm comprises acalculation of a projection associated with the first image data recordand a projection associated with the second image data record. Inparticular, the calculation of the projection associated with the secondimage data record comprises a coordinate transform on the basis of theorientation of the tilted object.

Expressed in formulae, a projectionproj₁ :=P _(rot1)(vol_(n))  (3)belonging to the first image data record and a projectionproj₂ :=P _(rot2)(vol_(n))  (4)belonging to the second image data record are calculated. Here, vol_(n)means the volume data record in the nth iteration step.

In a further preferred embodiment of the method according to theinvention, each pixel of the generated first image data record isdivided by the corresponding pixel of the projection associated with thefirst image data record, as result of which a modulated projection

${proj}_{1}^{*}:=\frac{{input}_{1}}{{proj}_{1}}$associated with the first image data record is obtained. Furthermore,each pixel of the generated second image data record is divided by acorresponding pixel of the projection associated with the second imagedata record, as a result of which a modulated projection

${proj}_{2}^{*}:=\frac{{input}_{2}}{{proj}_{2}}$associated with the second image data record is obtained.

In a further preferred embodiment of the method according to theinvention, a back projection, preferably an unfiltered back projection,is calculated on the basis of the modulated projection proj₁* associatedwith the first image data record and the modulated projection proj₂*associated with the second image data record.

In a further preferred embodiment of the method according to theinvention, the back projection is calculated as a sum of a backprojection, preferably an unfiltered back projection, associated withthe first image data record and a back projection, preferably anunfiltered back projection, associated with the second image datarecord.

Expressed in formulae, this back projection is calculated as follows:backproj:=P _(rot1) ^(T)(proj₁*)+P _(rot2)(proj₂*)  (5).

A further independent or alternative aspect for achieving the objectrelates to a method for generating an artefact-reduced 3D voxel datarecord of an object to be examined, with the aid of a computedtomography scanner, comprising the following steps in the specifiedsequence:

-   -   generating a first image data record by acquiring a multiplicity        of first computed tomography images of the object, wherein an        acquisition angle in respect of a first axis of rotation is        modified between the acquisition of the first computed        tomography images;    -   tilting the object by a predetermined tilt angle in respect of a        second axis of rotation which is arranged substantially        orthogonal to the first axis of rotation;    -   generating a second image data record by acquiring a        multiplicity of second computed tomography images of the object        tilted about the second axis of rotation;    -   reconstructing the first image data record in a first coordinate        system;    -   generating a second coordinate system by rotating the first        coordinate system on the basis of the orientation of the tilted        object;    -   reconstructing the second image data record in the second        coordinate system;    -   generating the voxel data record of the object to be examined,        by data fusion of the reconstructed image data records.

The explanations made above or below in respect of the embodiments ofthe first aspect also apply to the aforementioned further independent oralternative aspect and, in particular, to embodiments preferred in thisrespect. In particular, the explanations made above and below in respectof the embodiments of the respective other aspects in particular alsoapply to an independent aspect of the present invention and toembodiments preferred in this respect.

In accordance with the alternative aspect of the present invention, thereconstruction of the first image data record and of the second imagedata record respectively is carried out in a first coordinate system anda second coordinate system. Here, the second coordinate system emergesfrom the first coordinate system by rotating the first coordinate systemon the basis of the orientation of the tilted object. In particular, thesecond coordinate system emerges by rotating the first coordinate systemabout the predetermined tilt angle. Thus, the rotation is carried out,in particular, in such a way that the orientation of the reconstructedobject is substantially identical in respect of the first coordinatesystem and of the second coordinate system.

Within the meaning of this description, “data fusion” is understood tomean a combination of data, with the data fusion in particularcomprising an evaluation.

Two reconstructed image data records, i.e. two resultant voxel datarecords or volumes, are inherently aligned by means of the methodaccording to the invention. This makes the step of the data fusioneasier since no adaptation of the image data records in respect of theobject orientation is required anymore during the data fusion. The onlydifference between the values of the mutually corresponding voxels inthe two resultant volumes is either noise or an artefact. As a result ofa second coordinate system being generated by rotating the firstcoordinate system prior to the reconstruction of the second image datarecord according to the invention, wherein the second image data recordis reconstructed in said second coordinate system, the method accordingto the invention is only accompanied by a single interpolation step.Since each interpolation step is time intensive in view of the voxeldata record and, moreover, may be afflicted by errors, the methodaccording to the invention is superior in terms of speed and qualityover conventional methods, in which two or more interpolation steps arerequired.

In a preferred embodiment of the method according to the invention, thesecond coordinate system is obtained or calculated from the firstcoordinate system by means of the following transformation:G ₂(0,x ₂ ,y ₂ ,z ₂)=rot₂ ×G ₁(0,x ₂ ,y ₂ ,z ₂)×rot₂*  (6).

Here G₁ denotes the first coordinate system, G₂ denotes the secondcoordinate system, rot₂ denotes a rotation quaternion and rot₂* denotesthe rotation quaternion conjugate to rot₂.

The use of quaternions is simpler in handling compared to Eulerianangles and advantageously avoids the possibility of a gimbal lock.

In a further preferred embodiment of the method according to theinvention, the reconstruction of the first image data record and/or thesecond image data record is based on a back projection, preferably afiltered back projection. In particular, the reconstruction of the firstimage data record and/or the second image data record is carried out bymeans of a modified back projection, preferably a filtered backprojection.

However, in principle, it is also possible for the reconstructionalternatively to be based on an MLEM or for it to be carried out bymeans of an MLEM.

In a further preferred embodiment of the method according to theinvention, the modified back projection comprises a rotation of voxelcoordinates (x,y,z):(0,x′,y′,z′):=rot*·(ix+jy+kz)·rot  (7),where (x′,y′,z′) denote coordinates of the rotated coordinate system.The multiplications are quaternion multiplications in each case and rot*is the conjugate quaternion of rot.

In a further preferred embodiment of the method according to theinvention, the data fusion of the reconstructed image data recordscomprises an extremal value formation, i.e. a formation of a minimum ora formation of a maximum, of mutually corresponding voxels of the firstreconstructed image data record and the second reconstructed image datarecord.

In other words, the smallest or largest intensity value of the twomutually corresponding voxels of the first reconstructed image datarecord and the second reconstructed image data record is used for theresultant or fused voxel data record:vol_(ƒ)(x,y,z)=min{vol₁(x,y,z),vol₂(x,y,z)}  (8a),orvol_(ƒ)(x,y,z)=max{vol₁(x,y,z),vol₂(x,y,z)}  (8b).

In general terms, the data fusion can be carried out by means of afunction ƒ in a manner dependent on the reconstructed first image datarecord and the reconstructed second image data record:vol_(ƒ)(x,y,z)=ƒ{vol₁(x,y,z),vol₂(x,y,z)}  (8c).

In addition to forming the minimum and forming the maximum, thisfunction can also comprise other calculation operations, such as e.g.forming an average value. However, within the scope of the presentinvention, the extremal value formation, i.e. the formation of a minimumor a maximum, was surprisingly found to be particularly effective.

A further independent aspect for achieving the object relates to acomputer program product which comprises machine-readable program codewhich, when loaded onto a computer, is suitable for executing the methodaccording to the invention.

Below, individual embodiments for achieving the object are described inan exemplary manner on the basis of the figures. Here, the individualdescribed embodiments in part have features which are not mandatory forcarrying out the claimed subject matter, but which provide desiredproperties in specific cases of application. Thus, embodiments which donot have all features of the embodiments described below should beconsidered to be disclosed as falling under the described technicalteaching. Furthermore, certain features are only mentioned in relationto individual embodiments described below in order to avoid unnecessaryrepetition. Therefore, reference is made to the fact that the individualembodiments should be considered not only on their own, but also in anoverview. On the basis of this overview, a person skilled in the artwill identify that individual embodiments can also be modified byincluding individual features or a plurality of features from otherembodiments. Reference is made to the fact that a systematic combinationof the individual embodiments with individual features or with aplurality of features, which are described in relation to otherembodiments, may be desirable and expedient, and should therefore becontemplated and also be considered to be comprised by the description.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other objects, features, aspects, and advantage of the presentdisclosure will become better understood with regard to the followingdescription, claims, and drawings. The present disclosure is illustratedby way of example, and not limited by, the accompanying figures in whichlike numerals indicate similar elements. Moreover, a list of referencenumerals and corresponding explanations are provided in Table I.

FIG. 1 shows a schematic flowchart for reconstructing CT images inaccordance with the prior art;

FIG. 2 shows a schematic flowchart of the method according to theinvention in accordance with a preferred embodiment;

FIG. 3 shows a schematic sketch for acquiring a first image data record;

FIG. 4 shows a schematic sketch for acquiring a second image datarecord;

FIG. 5 shows a schematic flowchart of an image data reconstructionalgorithm for the method according to the invention in accordance with apreferred embodiment;

FIG. 6 shows schematic sketches for artefact reduction in accordancewith one example, in which two interpolation steps are necessary;

FIG. 7 shows schematic sketches for the artefact reduction according tothe invention in accordance with a preferred embodiment, in which onlyone interpolation step is required;

FIG. 8 shows photographic recordings of sectional images throughsynthetic 3D data of a test object, wherein the data of the image a) andthe data of the image b) differ by a 90° tilt of the object whengenerating the CT image data record;

FIG. 9 shows a photographic record of a sectional image throughsynthetic 3D data of the test object of FIG. 8 with reduced artefactsafter application of the method according to the invention;

FIG. 10 shows photographic recordings of sectional images throughphysical x-ray CT recordings of the test object, wherein image a) andimage b) differ by a 90° tilt of the object when acquiring the CTimages; and

FIG. 11 shows a photographic record of a sectional image throughphysical x-ray CT recordings of the test object from FIG. 10 withreduced artefacts after application of the method according to theinvention.

DETAILED DESCRIPTION

In the following description, use is made of the followingabbreviations, symbols and signs:

-   -   u and v denote the position of a pixel in a two-dimensional (2D)        image;    -   a is an index which specifies an image in an image sequence;    -   x, y and z describe the position of a voxel in a volume or voxel        data record;    -   n denotes the current iteration step;    -   input means an image sequence which was recorded by a computed        tomography scanner and which is used as input data record for        the MLEM;    -   input (u,v,a) describes an attenuation of x-ray radiation for        the pixel (u,v) in the image a;    -   input₁ is the image sequence which is recorded for generating a        first image data record; each image of this first image sequence        shows the whole object under a specific acquisition angle, with        the acquisition angle differing for various images in the first        image sequence;    -   input₂ is the image sequence which is recorded for generating a        second image data record; each image of this second image        sequence shows the whole object under a specific acquisition        angle, with the acquisition angle differing for various images        in the second image sequence;    -   vol₀ denotes the initial result volume or the result volume at        the outset;    -   vol_(n) denotes the result volume after the nth iteration step;    -   normseq, proj and proj* denote temporary image sequences;    -   backproj and backprojnorm denote temporary volume or voxel data        records;    -   norm is a normalization volume;    -   rot is a rotation in the form of a quaternion;    -   rot1 is a quaternion which describes the rotation of the object        for the first image data record, with rot1 therefore being an        identical rotation, i.e. rot1:=1, by definition;    -   rot2 is a quaternion which describes the rotation of the object        for the second image data record; to the extent that the tilt of        the object between the acquisition of the images of the first        image data record and the acquisition of the images of the        second image data record is 90° about the z-axis, the following        applies:

${{{rot}\; 2} = {\frac{1}{\sqrt{2}} + {k\frac{1}{\sqrt{2}}}}};$

-   -   P_(rot)(V) denotes an image sequence which is generated by a        forward projection of the volume V, which is rotated by the        quaternion rot    -   P_(rot) ^(T)(I) denotes a volume which is generated by an        unfiltered back projection of the image sequence I and the        inverse rotation rot;    -   P_(rot) ^(TF)(I) denotes a volume which is generated by a        filtered back projection of the image sequence I and the inverse        rotation rot;    -   R_(rot)(I) denotes a volume which is generated by a        reconstruction of the image sequence I and the inverse rotation        rot; to the extent that the filtered back projection is used as        a reconstruction algorithm, the following applies:        R_(rot)(I)=P_(rot) ^(TF)(I).

FIG. 1 schematically shows the dataflow as conventionally takes placewhen reconstructing CT images. CT reconstruction is understood to meanthe step characterized by reference sign in FIG. 1, in which a 3D voxeldata record or a 3D volume data record 4 is generated on the basis ofthe raw images recorded by a computed tomography scanner. Finally, the3D volume data record 4 describes the interior of the object to beexamined.

As depicted in FIG. 1, a multiplicity of CT images are initiallyrecorded under various perspectives or acquisition angles in a firststep 1 using the computed tomography scanner. Here, a series of imagesor an image sequence 2 is created which forms the starting point for thereconstruction 3. The reconstruction step 4 can be essentially carriedout using three different methods. The most commonly employedreconstruction methods are the so-called unfiltered and filtered backprojection. Alternatively, use is also made of iterative methods which,although they are more time-consuming, provide a higher resolution ofthe generated volume data record. One of these iterative methods is themaximum likelihood expectation maximization (MLEM) algorithm. After thereconstruction step 4, the volume data can be processed further,prepared or evaluated in a further step 5.

Step 6 depicted in FIG. 1, the so-called projection, simulates theinverse process of the step 3. Thus, an image sequence 2 is calculatedon the basis of a volume data record 4 during the projection. This stepis required, in particular, for the MLEM. The unfiltered back projectionis the transpose operation of the projection and it is used as part ofthe filtered back projection and as part of the MLEM.

The input data for the reconstruction process comprise an image sequenceacquired by the detector of the computed tomography scanner, or a seriesof images, with the series typically comprising approximately 1800images. Additionally, the input data also comprise metadata whichdescribe the position and recording angle of the object for each imagein the series. The results data of the reconstruction process comprise avoxel or volume data record which describes the attenuation of the x-rayradiation for each voxel of the object.

The methods of projection, of unfiltered back projection, of filteredback projection and of the MLEM algorithm, which are modified for themethod according to the invention, are described in more detail below.

Projection:

The projection is a process in which an image sequence is calculated onthe basis of a volume data record. Each image of the calculated imagesequence shows the result of a simulated x-ray image for a specificgeometry, i.e. for a specific acquisition angle, a specific objectposition and a specific distance between x-ray source and detector.

The projection proj=P_(rot)(vol) is calculated using the following stepsi) to iii), with the calculation taking place for all images a of theimage sequence and, in each image, for all pixels (u,v), where a∈{1, . .. , numImages}, with the number numImages of images in the series, andwhere (u,v)∈{1, . . . , numPixelU}×{1, . . . , numPixelV}, with thenumber numPixelU of pixels u and the number numPixelV of pixels v:

-   -   i) calculating the 3D coordinate point        (det_(x),det_(y),det_(z)), which corresponds to the detector        pixel (u,v), using the geometry or the meta data of the image a;    -   ii) rotating the coordinates of the x-ray source (source)        (src_(x),src_(y),src_(z)) and the coordinates of the detector        (det_(x),det_(y),det_(z)) in accordance with the inverse of rot:        (0,scr′_(x),scr′_(y),scr′_(z)):=rot*·(isrc_(x) +jsrc_(y)        +ksrc_(z))·rot  (9),        (0,det′_(x),det′_(y),det′_(z)):=rot*·(idet_(x) +jdet_(y)        +kdet_(z))·rot  (10),        where the multiplications are quaternion multiplications and        rot* is the conjugate quaternion of rot.    -   iii) calculating the line integral from the position of the        x-ray source (src_(x),src_(y),src_(z)) to the position of the        detector (det_(x),det_(y),det_(z)) by means of trilinear        interpolation and storing the result for the current pixel:

$\begin{matrix}{{{vec}:={\left( {\det_{x}^{\prime},\det_{y}^{\prime},\det_{z}^{\prime}} \right) - \left( {{scr}_{x}^{\prime},{scr}_{y}^{\prime},{scr}_{z}^{\prime}} \right)}},} & (11) \\{{{dist}:={{vec}}},} & (12) \\{{{dir}:=\frac{vec}{dist}},} & (13) \\{{{proj}\left( {u,v,a} \right)}:={\int_{0}^{dist}{{{vol}\left( {{src}^{\prime} + {s \cdot {dir}}} \right)}{{ds}.}}}} & (14)\end{matrix}$Unfiltered Back Projection:

The unfiltered back projection calculates a volume data record on thebasis of an image sequence. This operation is therefore the transposeoperation of the projection. The unfiltered back projection vol=P_(rot)^(T)(proj) is calculated on the basis of the following steps:

-   -   i) setting all voxels of the results data record vol to 0:        vol(x,y,z):=0  (15);    -   ii) carry out the following for all images a∈{1, . . . ,        numImages} and all voxels (x,y,z)∈{1, . . . , numVoxelX}×{1, . .        . , numVoxelY}×{1, . . . , numVoxelZ} of the results volume:        -   a) calculate the point (u,v) on the detector, on which a            line which passes through the x-ray source src and the point            (x,y,z) is incident (i.e. calculate the point of            intersection of the line with the detector plane); the            geometry or the metadata of the image a are used for the            calculation;        -   b) rotate the coordinates (x,y,z) in accordance with the            inverse of rot, see also Equation (7):            (0,x′,y′,z′):=rot*·(ix+jy+kz)·rot  (16),    -   where (x′,y′,z′) represent coordinates of the rotated coordinate        system. The multiplications are quaternion multiplications in        each case and rot* is the conjugate quaternion of rot.        -   c) add the value at (u,v) to the current value of the            results voxel using a bilinear interpolation, where the            value of 0 is used to the extent that (u,v) lies outside of            the input image:            vol(x′,y′,z′):=vol(x′,y′,z′)+proj(u,v,a)  (17).            Filtered Back Projection:

The above-described unfiltered back projection is disadvantageous inthat the resultant image is washed out and/or in that fine details arenot identifiable. Hence, a filtered back projection is usually used incomputed tomography, in which a digital filter, in particular ahigh-pass filter, is initially applied to the input data, before theunfiltered back projection, as described above, is carried out:P _(rot) ^(TF)(ƒ):=P _(rot) ^(T)(HighPass(ƒ))  (18).Maximum Likelihood Expectation Maximization (MLEM):

An alternative to the filtered back projection lies in iterativemethods, in which an initial estimate for the volume data record isimproved iteratively. Such iterative solutions have the advantage of alower noise and are therefore especially used in techniques such aspositron emission tomography, in which the signal-to-noise ratio is verylow. One iterative method is the MLEM. In MLEM, the problem of CTreconstruction is defined by means of a linear system of equations andsolved iteratively:A·vol=input  (19),where A represents a matrix which describes the projection operation,i.e. A·vol=P(vol).

The individual steps during the conventional MLEM reconstruction are asfollows:

-   -   i) calculating a normalization volume data record norm as an        unfiltered back projection of an image sequence, wherein all        pixels have a value of 1:        normseq(u,v,a):=1  (20),        norm:=P ^(T)(normseq)  (21);    -   ii) selecting an initial or inertial volume vol₀, wherein all        voxels are normally set to a value of 1, and setting the current        iteration index to 0:        vol₀(x,y,z):=1  (22),        n:=0  (23);    -   iii) calculating a projection of the current volume:        proj:=P(vol_(n))  (24);

How the projection is calculated was already explained above in thesection “Projection”.

-   -   iv) dividing each pixel in the input image sequence input by the        corresponding pixel in the image sequence proj of step iii):

$\begin{matrix}{{{{proj}^{*}\left( {u,v,a} \right)}:=\frac{{input}\left( {u,v,a} \right)}{{proj}\left( {u,v,a} \right)}};} & (25)\end{matrix}$

-   -   v) calculating the unfiltered back projection of proj*:        backproj:=P ^(T)(proj*)  (26);

How the unfiltered back projection is calculated was already explainedabove in the section “Unfiltered Back Projection”.

-   -   vi) dividing each voxel in backproj by the corresponding voxel        in the normalization volume:

$\begin{matrix}{{{{backprojnorm}\left( {x,y,z} \right)}:=\frac{{backproj}\left( {x,y,z} \right)}{{norm}\left( {x,y,z} \right)}};} & (27)\end{matrix}$

-   -   vii) setting each voxel of the results volume of the current        iteration step as results voxel of the preceding iteration step        multiplied by the corresponding voxel in backprojnorm:        vol_(n+1)(x,y,z)=vol_(n)(x,y,z)·backprojnorm(x,y,z)  (28);    -   viii) increasing the iteration index of the current iteration:        n:=n+1  (29);    -   ix) returning to step iii), provided n is less than the maximum        number of iteration steps.

What was identified within the scope of the present invention is thatthe MLEM algorithm can also be used for reducing artefacts by virtue ofthe algorithm being modified in such a way that it simultaneouslyprocesses a plurality of image data records, in particular two imagedata records.

Hence, in accordance with the invention, a first image data record and asecond image data record are recorded by way of the computed tomographyscanner, with the second image data record differing from the firstimage data record by a tilt of the object. In the modified MLEMalgorithm, all recorded images can be used and processed as input data.In other words, both the first and the second image data record are usedfor the reconstruction by means of the modified MLEM algorithm.

FIG. 2 shows a schematic flowchart of the method according to theinvention in accordance with a preferred embodiment. To this end, afirst image data record 10 is generated by virtue of a multiplicity offirst computed tomography images of the object to be examined beingacquired by means of a detector. Additionally, a second image datarecord 20 is generated after the object was tilted by a predeterminedtilt angle, in particular by 90°. Finally, an artefact-reduced 3D voxeldata record 40 of the object to be examined is calculated with the aidof an image data reconstruction algorithm 30 on the basis of thegenerated first image data record 10 and of the generated second imagedata record 20.

Thus, the object to be examined is scanned in the computed tomographyscanner with two different, in particular orthogonal, orientations. Theorientation of the 3-D object is advantageously described by means ofquaternions which, compared to the Eulerian angle, are much easier tohandle and moreover prevent the possibility of a “gimbal lock”. Aquaternion q encodes the object orientation in four numbers (a,b,c,d),of which a is the real part and (b,c,d) are the imaginary parts. Thequaternion is given by:q=a+ib+jc+kd  (30),wherei ² =j ² =k ² =ijk=−1  (31).

A conversion from an axis-angle representation, comprising an axisvector (ax+by+cz) and an angle θ, into a quaternion representation iscarried out as follows:

$\begin{matrix}{q = {{\cos\left( \frac{\theta}{2} \right)} + {{ia}\;{\sin\left( \frac{\theta}{2} \right)}} + {{jb}\;{\sin\left( \frac{\theta}{2} \right)}} + {{kc}\;{{\sin\left( \frac{\theta}{2} \right)}.}}}} & (32)\end{matrix}$

The basis orientation of an object, which represents no rotation, isgiven by the quaternion q=1. In this description, two orthogonalorientations are used, namely an orientation without rotation rot₁=1 andan orientation in which the object is rotated by 90° along the z-axis(0,0,1). The rotated orientation is given by the quaternion:

$\begin{matrix}{{rot}_{2} = {{{\cos\left( {45{^\circ}} \right)} + {k\;{\sin\left( {45{^\circ}} \right)}}} = {\frac{1}{\sqrt{2}} + {k{\frac{1}{\sqrt{2}}.}}}}} & (33)\end{matrix}$

For the purposes of reducing artefacts, CT recordings or CT scans of theobject are carried out in these orthogonal orientations rot₁ and rot₂.Hence, two image data records of projection images input₁ and input₂ areobtained. The respective projection images are linked with thecorresponding quaternion data in respect of the orientation in which theobject was scanned.

FIG. 3 shows a schematic sketch for acquiring the CT images of the firstimage data record, for which the object 80 to be examined is aligned inthe first orientation rot₁. FIG. 3 shows a section of a computedtomography scanner with an x-ray source 50 and a detector 70. The object80 to be examined, for example a printed circuit board, is situated on arotatably mounted object support 60. The object support 60 is arrangedbetween the x-ray source 50 and the detector 70 in such a way that theobject support 60 with the object 80 is rotatable through 360° about afirst axis of rotation. In FIG. 3, the first axis of rotation isoriented along the y-axis. For the purposes of generating the firstimage data record, the object is successively rotated about the firstaxis of rotation between the individually recorded images.

By way of example, the first image data record or the first imagesequence can comprise 1800 images by virtue of the object respectivelybeing rotated through 0.2° about the first axis of rotation after theacquisition of an image. In FIG. 3, the object 80 is arranged upright onthe object support 60, i.e. a longitudinal axis of the object 80indicated by the arrow is oriented along the first axis of rotation(y-direction).

FIG. 4 shows a schematic sketch for acquiring the CT images of thesecond image data record, for which the object 80 to be examined isaligned in the second orientation q₂. In contrast to the generation ofthe first image data record, as depicted in FIG. 3, the object 80 wastherefore tilted by 90° about a second axis of rotation. Here, thesecond axis of orientation is oriented in the z-direction, i.e.orthogonal to the first axis of rotation (y-direction). Analogous to thegeneration of the first image data record, the tilted object issuccessively rotated about the first axis of rotation between theindividually recorded images for the purposes of generating the secondimage data record. By way of example, the second image data record orthe second image sequence can also comprise 1800 images by virtue of theobject in each case being rotated by 0.2° about the first axis ofrotation after the acquisition of an image. In FIG. 4, the object 80 isarranged on its side on the object support 60, i.e. a longitudinal axisof the object 80 indicated by the arrow is oriented in the x-direction,i.e. orthogonal to the first axis of rotation and second axis ofrotation.

After the first image sequence and second image sequence were recorded,the results, i.e. the corresponding image data records, must bereconstructed and unified. Below, two alternative options are described,namely a reconstruction of both image data records by means of amodified MLEM algorithm and a reconstruction by means of filtered backprojection for the first image data record and the second image datarecord, and subsequent data fusion.

Reconstruction by Means of MLEM:

What was found within the scope of the present invention is that theMLEM algorithm, with the proviso that the above-described individualsteps must be at least partly modified or extended, is suitable for theimage data reconstruction algorithm 30 which can process both the firstimage data record and the second image data record to anartefact-reduced voxel data record in accordance with one aspect of theinvention. In particular, the MLEM algorithm needs to be modified insuch a way that both the first image data record and the second imagedata record can be used as input data for the algorithm.

The individual steps of a modified MLEM with artefact improvement are asfollows:

-   -   i) calculating a normalization volume data record norm as        unfiltered back projection of an image sequence, when all pixels        have a value of 1:        normseq₁(u,v,a):=1  (34),        normseq₂(u,v,a):=1  (35),        norm:=P _(rot1) ^(T)(normseq₁)+P _(rot2) ^(T)(normseq₂)  (36).

Here, in Equations (34) to (36) above, the index 1 relates to the firstimage data record and the index 2 relates to the second image datarecord. Correspondingly, normseq₁ refers to a normalized image sequenceof the first image data record and normseq2 refers to a normalized imagesequence of the second image data record.

-   -   ii) selecting an initial or inertial volume vol₀, wherein all        voxels are normally set to a value of 1, and setting the current        iteration index to 0        vol₀(x,y,z):=1  (37),        n:=0  (38);    -   iii) calculating projections of the current volume, see also        Formulae (3) and (4):        proj₁ :=P _(rot1)(vol_(n))  (39),        proj₂ :=P _(rot2)(vol_(n))  (40),

How the projections are calculated was already explained above in thesection “Projection”.

-   -   iv) dividing each pixel in the input image sequence input by the        corresponding pixel in the image sequence proj of step iii):

$\begin{matrix}{{{{proj}_{1}^{*}\left( {u,v,a} \right)}:=\frac{{input}_{1}\left( {u,v,a} \right)}{{proj}_{1}\left( {u,v,a} \right)}},} & (41) \\{{{{proj}_{2}^{*}\left( {u,v,a} \right)}:=\frac{{input}_{2}\left( {u,v,a} \right)}{{proj}_{2}\left( {u,v,a} \right)}};} & (42)\end{matrix}$

-   -   v) calculating the unfiltered back projection of proj*; see        Equation (5):        backproj:=P _(rot1) ^(T)(proj₁*)+P _(rot2) ^(T)(proj₂*)  (43);    -   How the unfiltered back projection is calculated was already        explained above in the section “Unfiltered Back Projection”.    -   vi) dividing each voxel in backproj by the corresponding voxel        in the normalization volume:

$\begin{matrix}{{{{backprojnorm}\left( {x,y,z} \right)}:=\frac{{backproj}\left( {x,y,z} \right)}{{norm}\left( {x,y,z} \right)}};} & (44)\end{matrix}$

-   -   vii) setting each voxel of the results volume of the current        iteration step as results voxel of the preceding iteration step        multiplied by the corresponding voxel in backprojnorm:        vol_(n+1)(x,y,z)=vol_(n)(x,y,z)·backprojnorm(x,y,z)  (45);    -   viii) increasing the iteration index of the current iteration:        n:=n+1  (46);    -   ix) returning to step iii), provided n is less than the maximum        number of iteration steps.

FIG. 5 shows the MLEM image data reconstruction algorithm in accordancewith a preferred embodiment on the basis of a schematic flow chart.Here, a volume or volume data record is denoted by a rectangle and animage sequence is denoted by an ellipse in each case. In step 100, thefirst image data record and the second image data record are provided asinput data 102. In step 101, a first estimator (e.g. 1) is initiallyassumed in order to calculate an initial volume data record 103.Finally, this volume data record is iteratively adapted or improved.Projections 105 are calculated on the basis of the volume data records103. The first input image data record and second input image datarecord 102 are divided by the result of these calculated projections 105in each case, as a result of which an image sequence ratio 104 isobtained. Back projections 108 are calculated in step 106. The result ofthese back projections 108 is divided by a first normalization volumedata record and a second normalization volume data record 109,respectively, which emerge from an unfiltered back projection 107,wherein a normalized volume data record 110 is obtained. Finally, instep 112, the output data are calculated for the next iteration step byvirtue of the normalized volume data record 110 being multiplied by thevolume data record 103 of the preceding iteration step. The result ofthis multiplication is the starting point for the next iteration step.

Reconstruction by Means of Filtered Back Projection and Subsequent DataFusion:

Filtered Back Projection:

The first image data record and the second image data record, i.e. theprojection data records input₁ and input₂, are reconstructed by amodified filtered back projection. The modification consists of thecoordinate system in which the object is reconstructed being rotated inaccordance with the object orientation prior to the actual backprojection.

For the first image data record or the projection data record input₁,the coordinate system, in which a filtered back projection is carriedout, is not rotated and represented by a first coordinate system orbasis coordinate system G₁.

For the second image data record or the projection data record input₂,the associated coordinate system, in which a filtered back projectionshould be carried out, is rotated equivalently to, or on the basis of,the object orientation rot₂, i.e. on the basis of the orientation of thetilted object, prior to carrying out the filtered back projection. Theangle about which the coordinate system is rotated thus corresponds tothe tilt angle about which the object was tilted in order to generatethe second image data record. A rotation (rot) of the first coordinatesystem G₁ into a second coordinate system G₂, in which finally thesecond image data record is reconstructed, can easily be carried out bypre-multiplication of each coordinate of the first coordinate system bythe rotation quaternion (rot) and post-multiplication by the conjugatequaternion (rot*); see also Equation (6):G ₂(0x ₂ ,y ₂ ,z ₂)=rot₂ ×G ₁(0,x ₁ ,y ₁ ,z ₁)×rot₂*  (47).

The first image data record is reconstructed in the first coordinatesystem G₁, as a result of which a first 3D voxel data record vol₁emerges, while the second image data record is reconstructed in thesecond coordinate system G₂, as a result of which a second 3D voxel datarecord vol₂ emerges:vol₁ =R _(rot1)(input₁)  (48),vol₂ =R _(rot2)(input₂)  (49),where R_(rotx) represents the reconstruction step and the inverserotation rotx.

A first advantage of the above-described procedure is that the tworesultant volume data records vol₁ and vol₂ are inherently aligned andthe only difference between the values of the mutually correspondingvoxels of the first volume and of the second volume is either noise oran artefact.

A second advantage is that, compared to conventional artefact reductionmethods, only a single interpolation step accompanies theabove-described procedure, and so the method according to the inventionis superior to the conventional methods in respect of speed and quality.

The second advantage of the method according to the invention emerges,in particular, from the fact that, according to the invention, thereconstruction of the second image data record is carried out in asecond coordinate system which is generated by rotating the firstcoordinate system or the basis coordinate system on the basis of theorientation of the tilted object prior to the reconstruction. In otherwords, the tilt of the object is already taken into account prior to thereconstruction of the second image data record. As a result, it ispossible to save an interpolation step. This will now be explained inslightly more detail on the basis of FIGS. 6 and 7:

While FIG. 6 shows schematic sketches for artefact reduction inaccordance with an example in which two interpolation steps arerequired, FIG. 7 shows schematic sketches for the artefact reductionaccording to the invention, in which, advantageously, only oneinterpolation step is required.

Images a to c of FIGS. 6 and 7 are identical. Here, Image a shows theobject to be examined in a first orientation. A first image data recordis generated in this orientation by acquiring first CT images. Theobject is reconstructed in a first coordinate system on the basis ofthis first image data record, as depicted in Image b. Image c shows theobject in a second orientation, namely tilted by 90° about the z-axis. Asecond image data record is generated in this orientation by acquiringsecond CT images. FIGS. 6 and 7 differ in terms of the procedure whichnow follows. While the reconstruction of the second image data record iscarried out in the same coordinate system in which the first image datarecord was also reconstructed in FIG. 6 (see FIG. 6d ), the coordinatesystem is rotated in accordance with the tilt of the object prior to thereconstruction of the second image data record in FIG. 7. Thereconstruction of the second image data record therefore takes place ina rotated or in a second coordinate system in FIG. 7. So that thereconstructed image data can be fused to one another, the coordinatesystem in which the second image data record was reconstructed must bealigned with the coordinate system in which the first image data recordwas reconstructed in accordance with the respective orientation of theobject such that the coordinates in each case reproduce the same pointsin the object. In the case of the procedure in accordance with FIG. 6,the object needs to be rotated in the coordinate system to this end (seeFIG. 6e ), while the object is rotated with the coordinate system in theprocedure in accordance with FIG. 7, which therefore requires no furtherreal calculation (see FIG. 7e ). Finally, FIGS. 6f and 7f indicate thedata fusion of the reconstructed first image data record and secondimage data record.

The substantial difference between the procedure in accordance with FIG.6 and the procedure according to the invention in accordance with FIG. 7is that, for the second volume, two interpolation steps, namely fromImage 6 c to Image 6 d and from Image 6 d to Image 6 e, are required inFIG. 6, while only one interpolation step, namely from Image 7 c toImage 7 d, accompanies the procedure in FIG. 7, i.e. the procedureaccording to the invention. In FIG. 7, the step from Image 7 d to Image7 e does not contain computational operations since the coordinatesystem is rotated together with the data. The second interpolation stepwas advantageously avoided by virtue of the rotation already beingintegrated into the reconstruction.

Data Fusion:

The reconstructed volume data records vol₁ and vol₂ were inherentlyaligned by means of the modified filtered back projection, i.e. thevoxel data in the two volume data records, which correspond to aspecific (x,y,z), represent the same point or region of the object to beexamined. All differences between mutually corresponding voxel data inthe two volume data records can therefore be identified as artefacts.Therefore, the following applies:vol₁(x,y,z)≈vol₂(x,y,z)  (50),andvol₁(x,y,z)−vol₂(x,y,z)=Δvol  (51),where Δvol represents the artefacts. As a result of the differentorientations in which the object was scanned, the metallic artefactstreaks in the two reconstructed volume data records vol₁ and vol₂ arenot situated at the same positions. The orthogonal orientations minimizethe possibility of overlapping metallic artefact streaks of the sameobject in vol₁ and vol₂.

The inventors found out that the artefacts bring about a positive ornegative deviation in the intensities from the actual or real value.Hence, a minimum data fusion algorithm or a maximum data fusionalgorithm is advantageous. To this end, the smallest or largestintensity value is used for the resultant or fused voxel data record oftwo mutually corresponding voxels of the first reconstructed image datarecord and the second reconstructed image data record (see alsoEquations 8a and 8b):vol_(ƒ)(x,y,z)=min{vol₁(x,y,z),vol₂(x,y,z)}  (52a),orvol_(ƒ)(x,y,z)=max{vol₁(x,y,z),vol₂(x,y,z)}  (52b).

The procedure in accordance with a preferred embodiment can therefore besummarized as follows:

-   -   calculating the filtered back projection of input₁ using rot₁=1        as a rotation (identity rotation):        vol₁ :=P _(rot1) ^(TF)(input₁)  (53);    -   calculating the filtered back projection of input₂ using rot₂ as        a rotation:        vol₂ :=P _(rot2) ^(TF)(input₂)  (54);    -   calculating the minimum or maximum of vol₁ and vol₂:        vol_(ƒ)(x,y,z)=min{vol₁(x,y,z),vol₂(x,y,z)}  (55a),        or        vol_(ƒ)(x,y,z)=max{vol₁(x,y,z),vol₂(x,y,z)}  (55b),        or

FIGS. 8 to 11 elucidate exemplary results of the artefact reduction ofthe method according to the invention. For the purposes of verifying theeffectiveness of the method according to the invention, a printedcircuit board with dimensions of 2.5 cm×2.5 cm was used as a testobject. The printed circuit board is a multi-layer board with six metallayers. Conventionally reconstructed 3D voxel data records, which arebased on CT scans of such a printed circuit board, generally havestreak-like or strip-like artefacts along metallic tracks, which make anaccurate detection of the track dimensions or of faults in the printedcircuit board more difficult or prevent this. Therefore, such a printedcircuit board is readily suitable as a test object. The method accordingto the invention was tested both using synthetically generated voxeldata records and physically generated x-ray computed tomography data.

FIG. 8 shows photographic recordings of sectional images throughsynthetic 3D data of the test object, with the data of image a) and thedata of image b) differing by a 90° tilt of the object when the CT imagedata record are generated. The synthetic 3D data were obtained on thebasis of the production data of the printed circuit board. Thecoordinate system was selected in such a way that the printed circuitboard is oriented parallel to the xy-plane. The synthetic data record ofthe test object is denoted by O^(S).

Initially, a forward projection of the synthetic data with thequaternion orientation rot₁=1 (i.e. no rotation) was carried out inorder to obtain the projection images P_(rot1)(O^(S)). These were thenreconstructed in a first grid G₁ (base grid) by means of filtered backprojection, as result of which the volume vol₁ ^(S) was obtained. Aslice through the xy-plane of vol₁ ^(S) is shown in FIG. 8A. It isvisible that the artefacts 200 of the metal tracks on the printedcircuit board extend in the horizontal direction, i.e. in thex-direction.

In a next step, O^(S) was rotated or tilted by 90° along the z-axis bymeans of the quaternion rot₂ (see Equation 33) in order to obtainO_(rot2) ^(S). A second forward projection on the basis of O_(rot2) ^(S)was carried out in order to obtain P_(rot2)(O^(S)). Prior to thereconstruction of these projection images, the second coordinate systemG₂, in which the back projection is undertaken, was generated byrotating G₁ through rot₂ in accordance with Equation (47). As a resultof this rotation, the reconstructed volume vol₂ ^(S) was inherentlyaligned with the reconstructed volume vol₁ ^(S). A slice through thexy-plane of vol₂ ^(S) is depicted in FIG. 8b . It is visible that, invol₂ ^(S), the position and direction of the artefact streaks 210 andcompletely different to in vol₁ ^(S) (FIG. 8a ). In vol₂ ^(S), theartefacts 210 do not extend in the horizontal direction as in vol₁ ^(S),but in the vertical direction, i.e. in the y-direction.

Finally, the volume data records vol₁ ^(S) and vol₂ ^(S) were fused byforming the minimum, like in the “Data Fusion” section above. A slicethrough the xy-plane of the volume data record vol_(ƒ) ^(S) areresulting therefrom is shown in FIG. 9. It is visible that the artefactsin vol_(ƒ) ^(S) are significantly reduced compared with vol₁ ^(S) andvol₂ ^(S) (see FIG. 8).

The method according to the invention was also tested on the basis ofphysically recorded x-ray computed tomography images. To this end, thetest object was successively scanned under two orthogonal objectorientations. For both scans, images were recorded at 800 differentacquisition angles in each case, with the resultant voxel dimensionsbeing 35 μm×35 μm×35 μm.

In the first scan, the printed circuit board was arranged in the CTscanner in such a way that it was aligned parallel to the flat paneldetector and therefore had the orientation rot₁. Projection images wererecorded in this orientation, as a result of which input₁ was obtained.In the second scan, the printed circuit board was rotated or tilted by90° along an axis oriented perpendicular to the flat panel detector,which axis intersects the x-ray source, in accordance with theorientation rot₂ in Equation 33. The projection data record obtainedthereby is denoted by input₂.

While input₁ was reconstructed by means of filtered back projection inthe first grid G₁ in order to obtain vol₁, input₂ was reconstructed bymeans of filtered back projection in a grid G₂ rotated in relation tothe first grid G₁ by rot₂ in order to obtain vol₂. Corresponding slicesthrough the xy-plane of vol₁ and vol₂ are shown in FIGS. 10a and 10b ineach case.

It is visible that both data records represent the same layer of theprinted circuit board and that, consequently, vol₁ and vol₂ areinherently aligned.

Like in the case of the synthetic data, which are presented in FIG. 8,there is a difference in the position and alignment of the artefactstreaks 200 for vol₁ and 210 for vol₂ in FIGS. 10a and 10b as a resultof the different object orientations when recording the data. Bothvolume data records vol₁ and vol₂ were fused using a maximum algorithm.

A corresponding slice through the xy-plane of the fused volume datarecord vol_(ƒ) is shown in FIG. 11. Here, it is once again visible thatthe artefacts could be significantly reduced. There are only still smallregions in vol_(ƒ) in which artefacts are still present, as can beidentified e.g. in the central left-hand and central upper regions ofFIG. 11. However, this can be traced back to inaccuracies when tiltingthe object by 90° since the tilt was carried out manually in theexperiments presented here.

TABLE 1 List of Reference Signs with Descriptors Reference NumeralDescription 1 Acquisition of CT images under different perspectives bymeans of CT 2 Image sequence 3 CT reconstruction/unfiltered or filteredback projection/MLEM 4 Voxel or volume data record 5 Further processing6 Projection 10 First image data record 20 Second image data record 30Image data reconstruction algorithm 40 Artefact-reduced 3D voxel orvolume data record 50 X-ray source 60 Object support 70 Detector/flatpanel detector 80 Object (e.g. printed circuit board) 100 Provision ofthe first image data record and the second image data record 101 Initialvalue (e.g. 1) 102 Input data record 103 Output or results volume datarecord 104 Input data record divided by projection/image sequence ratio105 Projection 106 Transposed projection 107 Transposed projection 108Back projection 109 Normalization volume 110 Back projection divided bynormalization volume/volume ratio 112 Output or results data record ofthe (n + 1)-th iteration step 200 Artefacts/horizontal artefact streaks210 Artefacts/vertical artefact streaks

We claim:
 1. A method for generating an artefact-reduced voxel datarecord of an object, the method comprising: generating, by a computingdevice, a first image data record by acquiring a multiplicity of firstcomputed tomography images of the object, wherein an acquisition anglein respect of a first axis of rotation is modified between theacquisition of the first computed tomography images; tilting the objectby a predetermined tilt angle in respect of a second axis of rotation,wherein the second axis of rotation is arranged substantially orthogonalto the first axis of rotation; generating, by the computing device, asecond image data record by acquiring a multiplicity of second computedtomography images of the object tilted about the second axis ofrotation; utilizing an iterative image data reconstruction algorithm togenerate the artefact-reduced voxel data record of the object based onthe generated first image data record and the generated second imagedata record, wherein the image data reconstruction algorithm comprises acalculation of a projection associated with the first image data recordand a projection associated with the second image data record, andwherein the calculation of the projection associated with the secondimage data record comprises a coordinate transform on the basis of anorientation of the tilted object; dividing each pixel of the generatedfirst image data record by the corresponding pixel of the projectionassociated with the first image data record; obtaining a modulatedprojection associated with the first image data record; dividing eachpixel of the generated second image data record by the correspondingpixel of the projection associated with the second image data record,and obtaining a modulated projection associated with the second imagedata record.
 2. The method according to claim 1, wherein the image datareconstruction algorithm is based on a maximum likelihood expectationmaximization algorithm.
 3. The method according to claim 1, wherein theimage data reconstruction algorithm comprises a calculation of anormalization volume data record as a sum of a normalization volume datarecord associated with the first image data record, and a normalizationvolume data record associated with the second image data record.
 4. Themethod according to claim 1, further comprising: calculating a backprojection based on the modulated projection associated with the firstimage data record and the modulated projection associated with thesecond image data record.
 5. The method according to claim 4, whereinthe back projection is calculated as a sum of a first back projectionassociated with the first image data record and a first back projectionassociated with the second image data record.
 6. A method for generatingan artefact-reduced voxel data record of an object, the methodcomprising: generating, by a computing device, a first image data recordby acquiring a multiplicity of first computed tomography images of theobject, wherein an acquisition angle in respect of a first axis ofrotation is modified between the acquisition of the first computedtomography images of the object; tilting the object by a predeterminedtilt angle in respect of a second axis of rotation, wherein the secondaxis of rotation is arranged substantially orthogonal to the first axisof rotation; generating, by the computing device, a second image datarecord by acquiring a multiplicity of second computed tomography imagesof the object tilted about the second axis of rotation; reconstructingthe first image data record in a first coordinate system; generating asecond coordinate system by rotating the first coordinate system on thebasis of an orientation of the tilted object, wherein the secondcoordinate system is generated from the first coordinate system byperforming the following transformation:G ₂(0,x ₂ ,y ₂ ,z ₂)=rot₂ ×G ₁(0,x ₂ ,y ₂ ,z ₂)×rot₂*, where G₁ denotesthe first coordinate system, G₂ denotes the second coordinate system,rot₂ denotes a rotation quaternion and rot₂* denotes the rotationquaternion conjugate to rot₂; reconstructing the second image datarecord in the second coordinate system; and generating, by the computingdevice, the voxel data record of the object based on data fusion of thereconstructed first image data record and the reconstructed second imagedata record.
 7. The method according to claim 6, wherein thereconstruction of at least one of the first image data record and thesecond image data record is based on a back projection.
 8. The methodaccording to claim 7, wherein the back projection comprises a rotationof voxel coordinates.
 9. The method according to claim 6, wherein thedata fusion of the reconstructed first image data record and thereconstructed second image data record comprises an extremal valueformation of mutually corresponding voxels of the first reconstructedimage data record and the second reconstructed image data record.
 10. Anon-transitory computer program product storing machine-readable programcode that, when executed, causes a computer at least to perform:generating a first image data record by acquiring a multiplicity offirst computed tomography images of an object, wherein an acquisitionangle in respect of a first axis of rotation is modified between theacquisition of the first computed tomography images; tilting the objectby a predetermined tilt angle in respect of a second axis of rotation,wherein the second axis of rotation is arranged substantially orthogonalto the first axis of rotation; generating a second image data record byacquiring a multiplicity of second computed tomography images of theobject tilted about the second axis of rotation; utilizing an iterativeimage data reconstruction algorithm to generate an artefact-reducedvoxel data record of the object based on the generated first image datarecord and the generated second image data record, wherein the imagedata reconstruction algorithm comprises a calculation of a projectionassociated with the first image data record and a projection associatedwith the second image data record, and wherein the calculation of theprojection associated with the second image data record comprises acoordinate transform on the basis of an orientation of the tiltedobject; dividing each pixel of the generated first image data record bythe corresponding pixel of the projection associated with the firstimage data record; obtaining a modulated projection associated with thefirst image data record; dividing each pixel of the generated secondimage data record by the corresponding pixel of the projectionassociated with the second image data record; and obtaining a modulatedprojection associated with the second image data record.
 11. Thenon-transitory computer program product recited in claim 10, wherein theimage data reconstruction algorithm is based on a maximum likelihoodexpectation maximization algorithm.
 12. The non-transitory computerprogram product recited in claim 10, wherein the image datareconstruction algorithm comprises: a calculation of a normalizationvolume data record as a sum of a normalization volume data recordassociated with the first image data record; and a normalization volumedata record associated with the second image data record.
 13. Thenon-transitory computer program product recited in claim 10, wherein theimage data reconstruction algorithm comprises a calculation of aprojection associated with the first image data record and a projectionassociated with the second image data record, and wherein thecalculation of the projection associated with the second image datarecord comprises a coordinate transform on the basis of an orientationof the tilted object.
 14. The non-transitory computer program productrecited in claim 13, wherein the machine-readable program code, whenexecuted, further causes the computer to: divide each pixel of thegenerated first image data record by the corresponding pixel of theprojection associated with the first image data record; obtain amodulated projection associated with the first image data record; divideeach pixel of the generated second image data record by thecorresponding pixel of the projection associated with the second imagedata record; and obtain a modulated projection associated with thesecond image data record.
 15. The non-transitory computer programproduct recited in claim 14, wherein the machine-readable program code,when executed, further causes the computer to: calculate a backprojection based on the modulated projection associated with the firstimage data record and the modulated projection associated with thesecond image data record.
 16. The non-transitory computer programproduct recited in claim 15, wherein the back projection is calculatedas a sum of a first back projection associated with the first image datarecord and a first back projection associated with the second image datarecord.
 17. A non-transitory computer program product storingmachine-readable program code that, when executed, causes the computerto: generate a first image data record by acquiring a multiplicity offirst computed tomography images of an object, wherein an acquisitionangle in respect of a first axis of rotation is modified between theacquisition of the first computed tomography images of the object; tiltthe object by a predetermined tilt angle in respect of a second axis ofrotation, wherein the second axis of rotation is arranged substantiallyorthogonal to the first axis of rotation; generate a second image datarecord by acquiring a multiplicity of second computed tomography imagesof the object tilted about the second axis of rotation; reconstruct thefirst image data record in a first coordinate system; generate a secondcoordinate system by rotating the first coordinate system on the basisof an orientation of the tilted object, wherein the second coordinatesystem is generated from the first coordinate system by performing thefollowing transformation:G ₂(0,x ₂ ,y ₂ ,z ₂)=rot₂ ×G ₁(0,x ₂ ,y ₂ ,z ₂)×rot₂*, where G₁ denotesthe first coordinate system, G₂ denotes the second coordinate systemrot₂ denotes a rotation quaternion and rot₂* denotes the rotationquaternion conjugate to rot₂; reconstruct the second image data recordin the second coordinate system; and generate an artefact-reduced voxeldata record of the object based on data fusion of the reconstructedfirst image data record and the reconstructed second image data record.